2 Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, IncThe Simplex MethodThe simplex method is an iterative process. Starting at some initial feasible solution (a corner point – usually the origin), each iteration moves to another corner point with an improved (or at least not worse) value of the objective function. Iteration stops when an optimal solution (if it exists) is found.Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc

3 Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, IncA Standard (maximization) Linear Programming Problem:The objective function is to be maximized.All the variables involved in the problem are nonnegative.Each constraint may be written so that the expression with the variables is less than or equal to a nonnegative constant.Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc

10 Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, IncThe Simplex Method:Set up the initial simplex tableau.If all entries in the last row are nonnegative then an optimal solution has been reached, go to step 4.Perform the pivot operation: convert pivot to a 1, then use row operations to make a unit column. Return to step 2.Determine the optimal solution(s). The value of the variable heading each unit column is given by the corresponding value in the column of constants. Variables heading the non-unit columns have value zero.Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc

11 Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, IncMultiple SolutionsThere are infinitely many solutions if and only if the last row to the right of the vertical line of the final simplex tableau has a zero in a non-unit column.No SolutionA linear programming problem will have no solution if the simplex method breaks down (ex. if at some stage there are no nonnegative ratios for computation).Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc

13 Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, IncA Standard (minimization) Linear Programming Problem:The objective function is to be minimized.All the variables involved in the problem are nonnegative.Each constraint may be written so that the expression with the variables is greater than or equal to a nonnegative constant.Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc

14 Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, IncThe Dual ProblemMaximization problems can be associated with minimization problems (and vice versa). The original problem is called the Primal and the associated problem is called the Dual.Theorem of DualityA primal problem has a solution if and only if the dual has a solution.Both objective functions attain the same optimal value.The optimal solution of the primal appears under the slack variables in the last row of the final simplex tableau associated with the dual.Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc

22 Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, IncIntroduce slack variables to make equations out of the inequalities and set the objective function = 0:The initial tableau and notice v = –2 (not feasible):We need to pivot to a feasible solutionCopyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc

23 Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, IncRatios82Locate any negative number in the constant column ( –2). Now go to the first negative to the left of that constant (–1). This determines the pivot column. The pivot row is found by examining the positive ratios. So –1 is our pivot.Create unit columnCopyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc

24 Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, IncNew pivot since it is the only positive ratioNote: now we have a feasible solution proceed with simplexCopyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc